Super-Exponential Complexity of Presburger Arithmetic

We consider the first-order theory whose language has as nonlogical symbols the constant symbols 0 and 1, the binary relation symbols = and This theory of integers under addition is commonly called the 'Presburger Arithmetic' and is known to be decidable for truth [Presburger (1929), Hilbert and Bernays (1968)]. We prove here that there exists a decision procedure for this theory, involving quantifier elimination, for which there is a superexponential upper bound on the size of formula produced when all variables have been eliminated.

Inspired by Boyer and Moore''s approach for generating induction schemes based on terminating function definitions, Zhang, Kapur and Krishnamoorthy introduced a cover set method for designing induction schemes for automating proofs by induction from specifications expressed as equations and conditional equations. This method has been implemented in the theorem prover Rewrite Rule Laboratory (RRL) and a proof management system Tecton built on top of em RRL, and it has been used to prove many nontrivial theorems and reason about sequential as well as parallel programs. The cover set method is based on the assumption that a function symbol is defined using a finite set of terminating (conditional or unconditional) rewrite rules. The left side of the rules are then used to design different cases of an induction scheme, and recursive calls to the function made in the right side can be used to design appropriate instantiations for generating induction hypotheses. A weakness of this method is that it relies on syntactic unification for generating an induction scheme for a conjecture. This paper goes a step further by proposing semantic analysis for generating an induction scheme for a conjecture from a cover set. We discuss the use of a decision procedure for Presburger arithmetic (quantifier-free theory of numbers with the addition operation and relational predicates , , =, ) for performing semantic analysis about numbers. The decision procedure is used to generate appropriate induction schemes for a conjecture using cover sets of function taking numbers as arguments. This extension of the cover set method automates proofs of many theorems which otherwise, require human guidance and hints. The effectiveness of the method is demonstrated using some examples which commonly arise in reasoning about specifications and programs. It is also shown how semantic analysis using a Presburger arithmetic decision procedure can be used for checking the completeness of a cover set of a function defined using operations such as check, many function definitions used in a proof of the prime factorization theorem stating that every number can be factored uniquely into prime factors, which had to be checked manually, can now be checked automatically in procedure for guiding generalization for generating conjectures and merging induction schemes is also illustrated.

Consider the first order theory of the real numbers with the function + (plus) and the predicate $ < $ (less than). Let S be the set of true sentences of this theory. We first present an elimination of quantifiers decision procedure for S, and then analyze it to show that it takes at most time $2^{2cn} $, where c is a constant, to decide sentences of length n. We next show that a given sentence does not change in truth value when each of the quantifiers is limited to range over an appropriately chosen finite set of rationals. This fact leads to a new decision procedure for S which uses at most space $2^{cn} $. We also remark that our methods lead to a decision procedure for Presburger arithmetic which operates within space $2^{2cn} $. These upper bounds should be compared with the results of Fischer and Rabin [2] that for some constant c, real addition requires time $2^{cn} $ and Presburger arithmetic requires time $2^{2cn} $.

Motivated by applications to program verification, we study a decision procedure for satisfiability in an expressive fragment of a theory of arrays, which is parameterized by the theories of the array elements. The decision procedure reduces satisfiability of a formula of the fragment to satisfiability of an equisatisfiable quantifier-free formula in the combined theory of equality with uninterpreted functions (EUF), Presburger arithmetic, and the element theories. This fragment allows a constrained use of universal quantification, so that one quantifier alternation is allowed, with some syntactic restrictions. It allows expressing, for example, that an assertion holds for all elements in a given index range, that two arrays are equal in a given range, or that an array is sorted. We demonstrate its expressiveness through applications to verification of sorting algorithms and parameterized systems. We also prove that satisfiability is undecidable for several natural extensions to the fragment. Finally, we describe our implementation in the πVC verifying compiler.

Decision procedures can be either theory-specific, e.g., Presburger arithmetic, or theory-generic, applying to an infinite number of user-definable theories. Variant satisfiability is a theory-generic procedure for quantifier-free satisfiability in the initial algebra of an order-sorted equational theory \((\varSigma ,E \cup B)\) under two conditions: (i) \(E \cup B\) has the finite variant property and B has a finitary unification algorithm; and (ii) \((\varSigma ,E \cup B)\) protects a constructor subtheory \((\varOmega ,E_{\varOmega } \cup B_{\varOmega })\) that is OS-compact. These conditions apply to many user-definable theories, but have a main limitation: they apply well to data structures, but often do not hold for user-definable predicates on such data structures. We present a theory-generic satisfiability decision procedure, and a prototype implementation, extending variant-based satisfiability to initial algebras with user-definable predicates under fairly general conditions.

Coq is a proof assistant featuring a tactic-based interactive theorem prover. The latest incarnation comes with over 150 tactics that assist the user in developing a formal proof. These tactics range from the simple and mundane to the ‘allpowerful’. Some examples from the latter category are the omega tactic that solves a goal in Presburger arithmetic and the ring and field tactics that solve identities modulo associativity and commutativity in ring and field structures. This paper presents a new proof tactic that decides equalities and inequalities between terms over lattices. It uses a decision procedure that is a variation on Whitman’s algorithm and is implemented using a technique known as proof by reflection. We will paint the essence of the approach in broad strokes and discuss the use of certified functional programs to aid the automation of formal reasoning. This paper makes three contributions. Firstly it serves an an introduction to using proof by reflection approach Coq. This utilizes the Ltac language and two recent extensions to the Coq system: type classes and the PROGRAM extension. Secondly it gives a certified implementation of Whitman’s algorithm in Coq, with proofs of correctness and termination. Thirdly, the final product of this work is a useable proof tactic that can be applied to proof goals involving equalities and inequalities in lattice theory. The rest of the paper is organized as follows: Section 2 will give a short introduction to Coq, highlighting the concept of a proof term, and will refresh the necessary details of lattice theory. We will give a motivation for the problem in Section 3, and a Coq-based introduction to free lattices along with the decision

This paper presents two decidability results on the validity checking problem for entailments of symbolic heaps in separation logic with Presburger arithmetic and arrays. The first result is for a system with arrays and existential quantifiers. The correctness of the decision procedure is proved under the condition that sizes of arrays in the succedent are not existentially quantified. This condition is different from that proposed by Brotherston et al. in 2017 and one of them does not imply the other. The main idea is a novel translation from an entailment of symbolic heaps into a formula in Presburger arithmetic. The second result is the decidability for a system with both arrays and lists. The key idea is to extend the unroll collapse technique proposed by Berdine et al. in 2005 to arrays and arithmetic as well as double-linked lists.Comment: arXiv admin note: text overlap with arXiv:1708.06696

This paper considers the satisfiability problem of symbolic heaps in separation logic with Presburger arithmetic and inductive definitions. First the system without any restrictions is proved to be undecidable. Secondly this paper proposes some syntactic restrictions for decidability. These restrictions are identified based on a new decidable subsystem of Presburger arithmetic with inductive definitions. In the subsystem of arithmetic, every inductively defined predicate represents an eventually periodic set and can be eliminated. The proposed system is quite general as it can handle the satisfiability of the arithmetical parts of fairly complex predicates such as sorted lists and AVL trees. Finally, we prove the decidability by presenting a decision procedure for symbolic heaps with the restricted inductive definitions and arithmetic.

Let T extend the theory P of Peano arithmetic, and suppose . Form from a model , in analogy to the way in which the ordered ring of integers is formed from the standard model of arithmetic. Let P′ and T′ be the corresponding analogues of P and T respectively. Now consider the group . In [5] MacDowell and Specker set out to determine the structure of such groups. (The precise statement in [5] refers to the ring of integers rather than the ordered ring. However, as pointed out to us by J. Knight, since Lagrange's Theorem that a positive integer is the sum of four squares is provable in the analogue of P′ for rings (see, for example, the proof in [7, p. 102]), the set of positive elements is definable in the ring, and consequently, so is the ordering. Thus, for the present purpose it makes no difference which of the two structures is used. Of course, one needs the ordering to discuss end extensions, as considered in [5]. On the other hand, one should be aware that in Pr′ one cannot define an ordering, where the theory Pr′ is the theory of the group of integers with distinguished element 1, 〈Z, +, 1〉. The constant 1 is needed so that divisibility mod n can be expressed. We will return to this point later.) In §1 we shall outline the results in this direction obtained in [5].Lipshitz and Nadel, unaware that a similar question had been posed and investigated in [5] (though, of course aware that [5] contained the celebrated results on end extensions) set out to characterize those models 〈A, +〉 of Pr = Presburger Arithmetic (the complete theory of 〈ω, +〉) which can be expanded to models 〈A, +, ·, 0, 1, ≤〉 of P. They were able to give a complete characterization for countable models 〈A, +〉 in [4], which we describe in §2.

This paper deals with model-checking of fragments and extensions of CTL* on infinite-state Presburger counter systems, where the states are vectors of integers and the transitions are determined by means of relations definable within Presburger arithmetic. We have identified a natural class of admissible counter systems (ACS) for which we show that the quantification over paths in CTL* can be simulated by quantification over tuples of natural numbers, eventually allowing translation of the whole Presburger-CTL* into Presburger arithmetic, thereby enabling effective model checking. We have provided evidence that our results are close to optimal with respect to the class of counter systems described above. Finally, we design a complete semi-algorithm to verify first-order LTL properties over trace-flattable counter systems, extending the previous underlying FAST semi-algorithm to verify reachability questions over flattable counter systems.KeywordsModel CheckTemporal LogicAtomic FormulaCounter SystemSymbolic Model CheckThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifier-free fragment, based on an encoding into the first-order theory of concatenation; the procedure has PSPACE complexity. The quantifier-free fragment of the theory of sequences can express properties such as sortedness and injectivity, as well as Boolean combinations of periodic and arithmetic facts relating the elements of the sequence and their positions (e.g., "for all even i's, the element at position i has value i+3 or 2i"). The resulting expressive power is orthogonal to that of the most expressive decidable logics for arrays. Some examples demonstrate that the fragment is also suitable to reason about sequence-manipulating programs within the standard framework of axiomatic semantics.

Some investigations into the algebraic constructive aspects of a decision procedure for various fragments of Relevant Logics are presented. Decidability of these fragments relies on S. Kripke's gentzenizations and on his combinatorial lemma known as Kripke's lemma that B. Meyer has shown equivalent to Dickson's lemma in number theory and to his own infinite divisor lemma, henceforth, Meyer's lemma or IDP. These investigations of the constructive aspects of the Kripke's-Meyer's decision procedure originate in the development of Paul Thistlewaite's “Kripke” theorem prover that had been devised to tackle the decision problem of the Relevant Logic R . A. Urquhart's pen and paper solution that relies on a sophisticated algebraic and geometric treatment of the problem shows the usefulness of an algebraic approach in Logic. Here, the study of the constructive aspects of the Kripke-Meyer decision procedure relies on various algebraic constructive results in the theory of polynomials rings.

Real or natural number interpretation and their effect on complexity

In this article we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P. Our research was motivated by the question whether one could study non-deterministic computation without loss of generality on a restricted, number theoretically significant class of nondeterministic Turing machines, the nondeterministic diophantine machines defined below [1,2]. The results suggest this is true.Because of the relative difficulty of the reduction of the satisfiability problem used in the proof, and the distinctly number-theoretic character of the problems shown to be NP-complete, we hope that the NP-completeness of these problems will play a role in showing the NP-completeness of further problems of a numerical nature, much as the satisfiability problem has in showing the NP-completeness of combinatorial problems ([3],[5]).The results illustrate an intimate connection between problems in computational theory, such as “P=NP?”, and problems in number theory, e.g. about quadratic congruences in one unknown, which are just beyond the range in which efficient algorithms exist. Thus our work exposes an interface between the state-of-the-art in number theory and in the theory of computation. Also, our results can be seen as the solution to a natural and important revision of Hilbert's 10thproblem: Give a feasible algorithmic procedure to decide whether an arbitrary diophantine equation has solutions. Unless P=NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact available. Certainly, these results present a striking rigorous demonstration that number theorists' intuitions about where problems about diophantine equations and quadratic congruences start to be truly difficult are justified.

Over the last decade, first-order constraints have been efficiently used in the artificial intelligence world to model many kinds of complex problems such as: scheduling, resource allocation, computer graphics and bio-informatics. Recently, a new property called decomposability has been introduced and many first-order theories have been proved to be decomposable: finite or infinite trees, rational and real numbers, linear dense order,...etc. A decision procedure in the form of five rewriting rules has also been developed. This latter can decide if a first-order formula without free variables is true or not in any decomposable theory. Unfortunately, this decision procedure is not enough when we want to express the solutions of a first-order constraint having free variables. These kind of problems are generally known as first-order constraint satisfaction problems. We present in this paper, not only a decision procedure but a full first-order constraint solver for decomposable theories. Our solver is given in the form of nine rewriting rules which transform any first-order constraint ? (which can possibly contain free variables) into an equivalent formula ? which is either the formula true, or the formula false or a simple solved formula having at least one free variable and being equivalent neither to true nor to false. We show the efficiency of our solver by solving complex first-order constraints over finite or infinite trees containing a huge number of imbricated quantifiers and negations and compare the performances with those obtained using the most recent and efficient dedicated solver for finite or infinite trees. This is the first full first-order constraint solver for any decomposable theory.